Transactions on Combinatorics
Volume:7 Issue: 1, Mar 2018

Let R be a commutative ring and M aný ýR-moduleý. ýIn this articleý, ýwe introduce a new generalization ofý ýthe annihilating-ideal graph of commutative rings to modulesý. ýTheý ýannihilating submodule graph of Mý, ýdenoted by G(M)ý, ýis aný ýundirected graph with vertex set A∗(M) and two distinctý ýelements N and K of A∗(M) are adjacent if N∗K=0ý. ýIný ýthis paper we show that G(M) is a connected graphý, ýýýdiam(G(M))≤3ý, ýand gr(G(M))≤4 if ýýG(M) contains a cycleý. ýMoreoverý, ýG(M) is an empty graphý ýif and only if ann(M) is a prime ideal of R and ýýA∗(M)≠S(M)∖{0} if and only if M is aý ýuniform R-moduleý, ýann(M) is a semi-prime ideal of Rý ýand A∗(M)≠S(M)∖{0}ý. ýFurthermoreý, ýRý ýis a field if and only if G(M) is a complete graphý, ýforý ýevery M∈R−Modý. ýIf R is a domainý, ýfor every divisibleý ýmodule M∈R−Modý, ýG(M) is a complete graph withý ýA∗(M)=S(M)∖{0}ý. ýAmong other thingsý, ýtheý ýproperties of a reduced R-module M are investigated whený ýG(M) is a bipartite graphý.

In this paperý, ýwe construct a new class of integral bipartite graphs (not necessarily trees) with large even diametersý. ýIn factý, ýfor every finite set A of positive integers of size k we construct an integral bipartite graph G of diameter 2k such that the set of positive eigenvalues of G is exactly A ý. ýThis class of integral bipartite graphs has never found beforeý.

The \emph{bipartite realisation problem} asks for a pair of non-negativeý, ýnon-increasing integer lists a:=(a1,…,an) and b:=(b1,…,bn′) if there is a labeled bipartite graph G(U,V,E) (no loops or multiple edges) such that each vertex ui∈U has degree ai and each vertex vi∈V degree bi. The Gale-Ryser theorem provides characterisations for the existence of a `realisation' G(U,V,E) that are strongly related to the concept of \emph{majorisation}ý. ýWe prove a generalisation; list pair (a,b) has more realisations than (a′,b), if a′ majorises a. Furthermoreý, ýwe give explicitly list pairs which possess the largest number of realisations under all (a,b) with fixed ný, ýn′ and m:=∑ni=1ai. We introduce the notion~\emph{minconvex list pairs} for themý. ýIf n and n′ divide m, minconvex list pairs turn in the special case of two constant lists a=(mn,…,mn) and b=(mn′,…,mn′).

In this paperý, ýwe obtain α as coefficient for the G=Kαn∪K(1−α)n¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ and by which we discuss Nikiforov's conjecture for λ1 and Aouchiche and Hansen's conjecture for q1 in Nordhaus-Gaddum type inequalitiesý. ýFurthermoreý, ýby the properties of the products of graphs we put forward a new approach to find some bounds of Nordhaus-Gaddum type inequalitiesý.

For any prime p let Cp(G) be the p-ary code spanned by the rows of the incidence matrix G of a graph Γý. ýLet Γ be the incidence graph of a flag-transitive symmetric design Dý. ýWe show that any flag-transitiveý ýautomorphism group of D can be used as a PD-set for full error correction for the linear code Cp(G)ý ý(with any information set)ý. ýIt follows that such codes derived from flag-transitive symmetric designs can beý ýdecoded using permutation decodingý. ýIn that way to each flag-transitive symmetric (vý,ýký,ýλ) design we associate a linear code of length vk that isý ýpermutation decodableý. ýPD-sets obtained in the described way are usually of large cardinalityý. ýBy studying codes arising from some flag-transitive symmetric designs we show that smaller PD-sets can be found forý ýspecific information setsý.